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Elliptic Tales: Curves, Counting, and Number Theory

Avner Ash and Robert Gross
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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What a fabulous book this is! Elliptic curves occupy a place of special importance in number theory, and are very much in the limelight these days. Their history is famous, of course, particularly regarding what transpired in the twentieth century.

An elliptic curve is the quintessential example of an abelian variety — and it’s wonderfully accessible to boot. It is cut out by a famous equation in two variables, y2 = a cubic in x, and is accordingly an algebraic variety (well, we really want to work projectively: it’s a projective variety). There’s a famous bit of trickery with lines meeting the curve that endows the points on the curve with an abelian group structure. The rational points on such a curve are in fact finitely generated: the Mordell theorem: see p. 152 of the book under review.

Of course, it is the work of Andrew Wiles in proving Fermat’s Last Theorem in the 1990s that has given the subject of elliptic curves its greatest boost in recent popularity, but it had been a favorite of arithmeticians and algebraic geometers for quite a while before that. And there is also its utility in cryptography: number theory as applied mathematics with a vengeance.

In any event, the subject of elliptic curves is beautiful, profound, important, and in fact accessible: witness its inclusion in so many courses at even the advanced undergraduate (or senior seminar) level. But now Ash and Gross have provided us with a new benchmark in the latter respect: says Swinnerton-Dyer, in his back-cover blurb, “Assuming only what every mathematically inclined freshman should know, this book leads the reader to an understanding of one of the most important conjectures in current number theory…”

This conjecture is nothing less than the famous conjecture of Birch and Swinnerton-Dyer himself: in the book under review we encounter it on p. 236 (of 248: good things come to him who waits) in the form stating that for any rational elliptic curve the analytic and algebraic rank coincide. What a beautiful conjecture! And then, a few pages later, as a finale, Ash and Gross address Tunnel’s result that the BSD conjecture implies “an explicitly computable criterion for whether or not n is a congruent number” and then show that n = 1234567 meets the bill! Truly exciting stuff.

All right, so it’s all about really wonderful mathematics. How about the book as such? Well, as I already indicated, it’s fabulous. It’s written in a largely conversational style, but without sacrificing rigor or slipping into triviality or vagueness. The book is not written in Satz-Beweis form — quite the contrary: after all, the theorems the authors deal with on their way to BSD are generally meant for insiders with a good deal of training and experience. The present audience, however, is quite different. The prerequisites Ash and Gross list for the book are, simply: “Algebra with coordinate geometry… Algebra with a bit of calculus… [and] A fair amount of calculus,” for three respective equivalence classes of chapters in the book. Instead, there are a lot of problems to be done by the reader, and examples are everywhere.

Thus, Elliptic Tales is geared toward leading an interested and sufficiently gifted kid setting out to learn some genuine number theory not by providing a network of proofs making for a sound theory (which will hopefully come later, when the kid’s in graduate school, doing number theory for real), but by showing him how things fit together. Ash and Gross do teach, or illustrate, a lot of material along the way, e.g., some appropriate group theory, what it means to count points, Dirichlet series, Taylor series, analytic continuation, L-functions, and so on: it couldn’t be otherwise. But they never lose sight of their goal, i.e. p. 236.

Finally, so as not to misrepresent the material even a little, let me note also that, for example, the fifth chapter in the book, on Bézout’s Theorem, does include a marvelous sketch of its proof. Ash and Gross then follow the sketch with an “illuminating example,” which of course fits with their overarching pedagogical objective. Manifestly, the authors are explicitly preparing their charges for true mathematics, replete with all its dotted I’s and crossed t’s, even if they can’t hit the heavy proofs dead-on yet.

I cannot imagine a better book to use with the right group of gifted rookies. Elliptic Tales is a fabulous book.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

The table of contents is not available.